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- 1. Cantor and Infinity
- 2. Numbers and Infinite Numbers We encounter problems when we start asking questions like: − What is a number? − What is infinity? − Is infinity a number? − If it is, can there be many infinite numbers? We can resolve some aspects of these questions with a few simple ideas, rigorously applied.
- 3. Some Definitions A set is a collection of well-defined, well- distinguished objects. These objects are then called the elements of the set. For a given set S, the number of elements of S, denoted by |S|, is called the cardinal number, or cardinality, of S. A set is called finite if its cardinality is a finite nonnegative integer. Otherwise, the set is said to be infinite.
- 4. Galileo’s Paradox of Equinumerosity Consider the set of natural numbers Ν = {1, 2, 3, 4, …} and the set of perfect squares (i.e. the squares of the naturals) S = {1, 4, 9, 16, 25, …}. Galileo produced the following contradictory statements regarding these two sets ...
- 5. A Contradiction? 1.While some natural numbers are perfect squares, some are clearly not. Hence the set N must be more numerous than the set S, or |N| > |S|. 2.Since for every perfect square there is exactly one natural that is its square root, and for every natural there is exactly one perfect square, it follows that S and N are equinumerous, or |N| = |S|.
- 6. Many Contradictions? We could repeat this reasoning for N and: − The even numbers E = {2, 4, 6, …} − Triples − Cubes − Etc. In each case we have systematically picked out an infinite proper subset of N. A is a proper subset of B if A is contained in B, but A ≠ B.
- 7. One-to-One Correspondence Galileo’s exact matching of the naturals with the perfect squares constitutes an early use of a one-to-one correspondence between sets – the conceptual basis for Cantor’s approach to infinity.
- 8. Galileo's 'Solution' To resolve the paradox, Galileo concluded that the concepts of “less,” “equal,” and “greater” were inapplicable to the cardinalities of infinite sets such as S and N, and could only be applied to finite sets. Cantor showed how these concepts could be applied consistently, in a theory of the properties of infinite sets.
- 9. The Basis of Cantor's Theory A bijection is a function giving an exact pairing of the elements of two sets. Two sets A and B are said to be in a one-to- one correspondence if and only if there exists a bijection between the two sets. We then write A~B. A set is said to be infinite if and only if it can be placed in a one-to-one correspondence with a proper subset of itself.
- 10. Which sets have the same cardinality as N? The cardinality of the natural numbers, |N|, is usually written as Other well-known infinite sets have cardinality − The integers, I, { …, -3, -2, -1, 0, 1, 2, 3, …} (fairly easy to show N~I). − The rationals, Q (a bit more difficult to show N~Q). Showing N~I
- 11. Showing N~Q We can establish a one-to-one correspondence between the naturals and the rationals. We use an infinite 2- d arrangement of Q, and a systematic path through this arrangement.
- 12. Are there any infinite sets with greater cardinality than N? We can show that |R| is greater |N|. We cannot establish a one-to-one correspondence between N and R. |R| is a higher order of infinity, usually written as Let's prove this ...
- 13. Cantor's Diagonal Argument For any hypothesised enumeration of the real numbers, we can show that there is a real which is not in that enumeration. We rely on forming a new real by the systematic alteration of the digits in the enumeration.
- 14. Transfinite Arithmetic Cantor devised a new type of arithmetic for these infinite numbers. For the infinite numbers we have rules such as these ... Note: the cardinality of the set of real numbers is often written 'c' for 'continuum'.
- 15. The Continuum Hypothesis We have seen that |R| is greater than |N|. But, are there any infinite numbers in between these? The hypothesis that there is no infinite number between |N| and |R| is called the continuum hypothesis. It was shown to be formally undecidable by Gödel and Cohen.

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